The symmetric crosscap number of the families of groups \(DC_{3}\times C_{n}\) and \(A_{4}\times C_{n}\)
type: Item
instance of: scholarly article
zbMATH Open document ID: 1252.57008
publication date: 2012-07-31T00:00:00Z
full work available at URL: Vol38-2.html
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 14H37
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 20F38
Mathematics Subject Classification ID: 20H10
zbMATH DE Number: 6062410
zbMATH Keywords: symmetric crosscap number
zbMATH Keywords: non-orientable surface
zbMATH Keywords: group action
author: José Javier Etayo
author: Ernesto Martínez
MaRDI profile type: Publication
title: The symmetric crosscap number of the families of groups \(DC_{3}\times C_{n}\) and \(A_{4}\times C_{n}\)
published in: Houston Journal of Mathematics
review text: Every finite group \(G\) acts faithfully as a group of automorphisms of some non-orientable Klein surface without boundary. The minimum genus of all such surfaces is called the \textit{symmetric cross-cap number} of \(G\), and denoted by \(\tilde{\sigma}(G)\). This parameter is known for several classes of finite groups and for all groups of order up to 127 (following work by Coy May, the authors of this paper, and the reviewer).NEWLINENEWLINESome recent attention has been paid to the inverse problem of determining which positive integers can occur as values of \(\tilde{\sigma}\). It was proved by \textit{C. L. May} [Glasg. Math. J. 43, No.3, 399-410 (2001; Zbl 0997.57034)] that there are groups with symmetric cross-cap number 1, 2 and any integer from 4 to 16 inclusive, but no finite group \(G\) with \(\tilde{\sigma}(G) = 3\). Subsequently the authors of this paper proved in [Proc. R. Soc. Edinb., Sect. A, Math. 138, No. 6, 1197-1213 (2008; Zbl 1165.30022)] that every integer \(g > 10\) such that \(g \equiv 0, 1\) or \(2\) mod \(4\) is the symmetric cross-cap number of \(C_m \times D_n\) for some \(m\) and \(n\), and that for every positive integer \(k\), the group \(C_{6k} \times C_3\) has symmetric cross-cap number \(12k-1\) (\(\equiv 11\) mod \(12\)). Also the reviewer has proved (in unpublished work) that for every positive integer \(k\), there is a semi-direct product \(C_n \rtimes S_4\) with symmetric cross-cap number \(3k-2\) (\(\equiv 1\) mod \(3\)). It follows that every positive integer not congruent to \(3\) mod \(12\) is now known to be \(\tilde{\sigma}(G)\) for some \(G\).NEWLINENEWLINEIn this paper, the authors determine the symmetric cross-cap numbers of the groups \(DC_3 \times C_n\) (where \(DC_3\) is the dicyclic group \(C_3 \rtimes_{-1} C_4\) of order \(12\)) and \(A_4 \times C_n\), for all \(n\). In doing so, they prove that these numbers cover a quarter of all positive integers congruent to \(3\) mod \(12\) (and three quarters of of all positiv…
reviewed by: Marston D. E. Conder
description: scientific article; zbMATH DE number 6062410